Abstract: The concept of Fourier synthesis provides the foundation for both everyday
consumer electronic products and fundamental research. In the latter, so called
pulse shaping is nowadays key to dynamically initialize, probe and manipulate
the state of classical or quantum systems. In nuclear magnetic resonance, for
instance, shaped pulses have a long-standing tradition and the underlying
fundamental concepts have subsequently been successfully extended to optical
frequencies and even to implement quantum gate operations. Transfer of these
paradigms to nanomechanical systems, requires tailored nanomechanical waveforms
(NMWFs) for mechanically mediated coherent control schemes. Here, we report on
a novel additive Fourier synthesizer for NMWFs based on monochromatic surface
acoustic waves (SAWs). As a proof of concept, we electrically synthesize four
different elementary NMWFs from a fundamental SAW at $ f_1 \sim 150$ MHz using
a superposition of up to three discrete harmonics $f_n$. We employ these shaped
pulses to interact with an individual sensor quantum dot (QD) and detect their
deliberately and temporally modulated strain component via the opto-mechanical
QD response. As a very attractive feature and in contrast to the commonly
applied direct mechanical actuation by bulk piezoactuators, SAWs provide more
than two orders of magnitude larger frequencies > 20 GHz to resonantly drive
mechanical motion. Thus, our technique uniquely allows coherent mechanical
control of localized vibronic modes of optomechanical and phoXonic crystals,
most tantalizing even in the quantum limit when cooled to the vibrational
ground state.